Abstract
A method to obtain the orientation distribution function (ODF) of a polycrystalline material directly from X-ray diffraction spectra is presented. It uses the maximumtexture-entropy assumption to reduce the diffraction data needed for the ODF analysis. The validity of this new method is illustrated through two model examples.

spectra. Its validity is illustrated through two simulated examples.

2. Mathematical algorithm
Let us consider a single-phase polycrystalline sample whose crystal structure is known. In the case of a random grain-orientation distribution, the integrated intensity, l~l, of the {h} Bragg diffractions is given by the kinematic approximation (Taylor, 1961). R = l{h}
with

I. Introduction
The orientation distribution function (ODF) of a polycrystalline material is generally determined from polefigure data by the various pole-figure inversion methods such as the harmonic method (HM) (Bunge, 1965; Roe, 1965), the vector method (VM) (Ruer & Baro, 1977; Vadon, 1981) and the WIMV method (Matthies & Vinel, 1982). Recently, X-ray (or neutron) diffraction spectra have been used to determine the ODFs of some polycrystalline materials (Wenk, Matthies & Lutterotti, 1994). Such an approach is expected to provide not only an efficient means for the quantitative texture analysis of complex materials (such as intermetallics, ceramics and duplex phase alloys) with single or/and overlapping diffraction peaks, but also a significant tool for the fast texture measurements of traditional materials. In the last decade, the maximum-entropy concept has been applied to the ODF determination from pole-figure data for cubic, hexagonal and tetragonal materials (Wang, Xu & Liang, 1987; Schaeben, 1988). It has been demonstrated that fewer pole-figure data may be required by the use of a refined algorithm, i.e. the socalled modified maximum-entropy method (MMEM) (Wang, Xu & Liang, 1996). The new algorithm introduces directly the maximum-entropy concept into the least-squares-equation of the pole-figure inversion. Thus, it is possible to perform the partial ODF analysis with only a limited number of input pole-figure data normalized according to a standard textureless sample (Wang, Vadon, Heizmann & Xu, 1996). In the present paper the MMEM is extended to the determination of the complete ODFs directly from X-ray diffraction
@, 1997 International Union o f Crystallography Printed in Great Britain - all rights reserved

KS(h)

(1)

S(h)

=

P(O)N{h } IF{h}12exp(-2B sin 2 0/A2),

(2)

where K is a constant, P(O) is the Lorentz polarization factor, which is a function of the Bragg diffraction angle 0, F{h} is the theoretical structure factor, N{h} is the multiplicity factor, B is the Debye-Waller factor of the measured sample and 2 is the wavelength of the incident X-rays. For a textured material the integrated intensity, I~}, of the {h} Bragg diffraction at a certain sample direction y must be expressed as (Hedel, Bunge & Reck, 1994) I{~I(Y) = l~h}P{hl(Y)' (3)

where the ~,,,,, are the matrices relating the texture coefficients Wl,,,, and f(gj) and t h e elm,,(y {h} ) are the
Journal of Applied C~stallography
ISSN 0021-8898 ~(',~ 1997

(6)
VT[K(°)' ~(i)] iiVT[K<O). (iii) after several iteration steps. In general.m.m. 1981). k is approached by the iterative processes
k(i+l) = ~(i) __ O~
matrices relating Wlmn and the pole-figure densities. and the P'~(x) are the associated Legendre functions.n.
(12)
/irl(y) = KS(h)
x ~
j=l I' max
/max 1' l' y~ y ~ ~ g'lmn(y){h} 1=0 m=-l n=-l
where i[~]r(y) are the recalculated intensities of some diffraction peaks in the ith iteration..O~.. r/) and (O. 1988) is suggested. I~I(Y).1-2o
X
3. = 25 °
~= 5°
. Model examples
(8)
-.
7.OJl..y
II~I(Y)-.."
i . no defocusing corrections to the diffraction intensities would be required if the Schulz reflection method is used to obtain X-ray diffraction data... The lma x value in (8) should correspond to the sharpness of texture.y ~(i+1) = ~(i) __ ~ h.. q)) are the spherical polar coordinates of y and h."'"° °. Wenk. 1996)...Z
l'=1 m'=-l' n'=-l'
Z
Z
21.KS(h) y]
Z
I=0 m=-I n=-I
y] e. ~-(i)]
IIV T[K (i+~).. Here. respectively•
dZm...: i .. As the polar angles Z of the input diffraction spectra are not larger than 50 °.(Y) read Ih) £lmn(Y) : 2rt[2/(21 + 1)]l/2p~/(cos Z) x exp (-imr/)/~/(cos 69) exp (indP). the ODF may be written as (Wang.. .lmn(Y Ihl )
±
J=l l' max 1' 1' ) ]2 = min.y
(7)
where 2o and 2t. the selection of lmax in (8) depends mainly on the total number of the input I~}(y) used for the ODF analysis and the crystal structure of the measured sample.
: . of several {h} Bragg diffraction peaks at the different sample directions y are measured. while K is kept fixed.m. It proceeds as follows: (i) the initial values of K. Xu & Liang. In each case a model ODF is constructed with the certain Gaussian peak-type texture components (Wagner.. . Introducing (4)-(7) into (2) results in
VT[ K(i+l).. Xu & Liang. In this paper lma x and l~ax are chosen to be 22 and 10. k (°) = 0
Al. are set as K(°) = Y~ /~I(Y)/Y~ S(h) h.y .n. iiiii!ii.n. [I~h))r(Y)]2
h. if the real intensities.y h. 1..5"
where k is the vector expression of unknown 2 o and
Fig. I.-. Xu & Liang.. a new iterative process is used
(
-1-20-Y
]
1'=1 ra'=-l' n=-l'
y]
~
21.
)
K(i+]) = E l~l(Y)I~h~r(Y)/ ~-..= 50 °
(10)
7.-.ex p
1' I'
.m.. {Ill etm. respectively.-"° I
r 1 = 67. 1987) f(gy)=exp
the optimal step length 0e is determined in each iteration so as to minimize the objective function T. the Lagrangian multipliers in (7) can be derived by solution of the following least-squares equation:
lmax 1 l
T : Y]
h. Esling & Bunge.. all the unknown multipliers are to be evaluated directly from the X-ray diffraction spectra...!iii I" ..
-. m.n...n.Z
Z Z 2l'm'n'O~'m'n ' 1'=1 m'=-l' n=l'
(9)
'~'5° o
To obtain a convergent solution of the normalized factor K and the unknown multipliers 2t. 2 o and 21'm'. k (°] II
. For the 15 modelled spectra the corresponding sample directions are shown in Fig. On the assumption of maximum texture entropy.m..
The method described above is tested by the two model examples: for the case of cubic-orthorhombic symmetry and for that of tetragonal-orthorhombic symmetry.. the gradient method (Brousse. ° " .m.
Thus. Illustration of the positions of 15 measuring points on the pole sphere selected for the ODF analysis.444
QUANTITATIVE TEXTURE ANALYSIS FROM X-RAY DIFFRACTION SPECTRA (ii) in the first iterative stage. 1987) or directly from the pole-figure inversion equation (Wang.k(i)]l I
(11)
where (Z..
.. A total number of 15 X-ray diffraction spectra are predetermined by the HM (/max = 22).. are the unknown Lagrangian multipliers that can be determined either from the texture coefficients (Wang.

by the present method and the classic HM.
3. This is due to the fact that in the classic HM no positivity condition is involved and the use of fewer diffraction data generally leads to a smaller number of determinable texture coefficients. The comparison between the modelled {200} pole figure (Fig. 7. 4. the much lower orientation intensities at the peak positions of main texture components are obviously observed. The accuracy criterion of recalculated intensity is given by the parameter R (Ruer & Baro. are summarized in Table 1. {222}. D. It is seen that the ODF reproduced from the diffraction spectra is in very good agreement with the model one. 1977). 13.)
o
D
((')
l0 ~11
Fig. 13. 2. Fig..
(13)
h. The same sections obtained by the present method from the 15 X-ray diffraction spectra are displayed in Fig. 5. WANG et al. composed of {111 }(112) and {110} (110) texture components.
R (h'~[/l y (Y)~ ~[l~-I~-~Y~-/I~. the assumed diffraction spectra consist of five single peaks. 2(a).
. {200}. 10 ° . 2(c) gives the constant q~ sections of the ODF that is obtained from the same diffraction spectra by the classic
HM. i. 6.]'
b
\
b
0
b
b
)
>
D
09
b@
\
b
(a) (b)
l O ~i
}
}
)
.(y)]2~/
=
1/2
x 100%.) sections of the ODFs (cubic-orthorhombic symmetry) (a) modelled by the {111}(112) and {110}(110) Gaussian peak-type texture components with the full width at half-maximum (FWHM) of 15 ° (levels: !.1. 10).Y. 8. {311} and {511 } (the powder diffraction intensities are input according to the ASTM card of Cu sample). The constant ~o(0. 3a) and that recalculated from the derived ODF (Fig 3b) confirms that it is possible to obtain quantitative texture information from the diffraction spectra. {220}. compared with the model ODF. It is evident that. 16) (b) calculated by the present method (levels: 1.. 2(b). 4. The constant ¢p sections of the model ODF.y
. 4. The detailed comparisons between the numerical results of the ODF analysis from diffraction spectra. 16) and (c) calculated by the classic HM (levels: 2. Cubic-orthorhombic symmetry
445
For the cubic-orthorhombic symmetry case. are shown in Fig.e. 7.

3. i. for the present method. 5. This suggests that the ODF may also be determined directly from those diffraction spectra with some overlapping peaks. when the ODF is determined directly from the incomplete pole figures. 4(b). the quantitative relationships among the different {h} diffraction planes. As can be seen. the classic HM produces the relatively lower orientation densities at the peak positions and the smaller volume fractions of the main texture components. usually more input diffraction data with the polar angle larger than 50 ° are required (Wang. (201). Further comparisons between the numerical results of the ODF analysis by our new algorithm and the classic HM are given in Table 2.446
QUANTITATIVE TEXTURE ANALYSIS FROM X-RAY DIFFRACTION SPECTRA tation densities at the peak position and the volume fractions of the two main texture components produced by the new algorithm are very close to the model values. respectively. the often-
3. its ~ value is obviously larger. r / = 0 ° and at Z = 50°. are used as the input data for the ODF analysis. 15 simulated 'experimental' diffraction spectra. i. are very small.90 °.
(14)
where the sum is taken over all the range of the Euler space andfm(gj) andfC(gj) are the model and calculated orientation densities at certain orientation gj.796
(b) Fig. 1. r / . According to the geometric set in Fig. (001). (110). In contrast. Xu & Liang. 1993). Here. 3. are introduced into the texture analysis. (002)+(200). the volume fraction . Muller & Esling. Tetragonal-orthorhombic symmetry We refer to the powder X-ray diffraction data of TiA1 alloy with LIo structure (Wang. Fig. It indicates that it is possible to perform a quantitative ODF analysis with a limited number of diffraction spectra for tetragonal materials. Levels 1. 4(a) gives the constant tp sections of the model ODF. 1996). 2(b).4 V/V(tni) of the two main texture components are also listed in Table 1.
. 4. The obvious reduction of the input diffraction data in this example comes from the fact that more constraints. whereas relatively larger R and 6 values and considerably smaller volume fractions of the main texture components compared with the model texture are associated with the classic HM.e. There is good agreement between the two ODFs.4 V/V(ogi) is defined as an integral intensity of orientation densities about the peak region up to a fixed orientation distance coi from the peak position gi (Zuo. good agreement between the simulated and recalculated spectra with the rather small residual errors of their diffraction intensities for the single and overlapping diffraction peaks is evident.e. Fig. Sun. {200}pole figure (a) modelled and (b) recalculated from Fig. However. It should be mentioned that the accuracy of the ODF analysis is
'•RD
66337
L
(a) ~RI)
~ ~ M a x : 6. The relative orientation densities at the peak positions and the volume fractions . The same sections of the ODF obtained from the 15 X-ray diffraction spectra by the present method are displayed in Fig. 2. (111). The model texture is constructed with the two peaktype Gaussian components. each consisting of seven single or/and overlapping peaks. (111)[11"2] and (110)[110]. (202)+(220) and (113) + (311). As shown in Table 1. 6.2. Also. Chen& He. The above two examples indicate that our algorithm may allow the quantitative texture information to be obtained directly from the diffraction spectra. both the R value and the 6 value. 1990)
3=
~_
[fm(gj)_fC(gj)]2
. 1992). As compared with the model ODF. in spite of there being almost the same R value for the classic HM. Satisfactory results have been observed for the present method. 5 shows the simulated and recalculated diffraction spectra at X = 50°.
The average error between the model and recalculated ODFs is defined as (Wright & Adams.